Bernstein Polynomials Research Articles

Optimization of the Progressive Image Mosaicing Algorithm in Fine Art Image Fusion for Virtual Reality

The fade-in and fade-out algorithm based on the Bernstein polynomial has certain limitations in image fusion. Therefore, this article proposes a new image fusion algorithm. First, the SIFT algorithm is used to register the images. Second, for the disjointed case of overlapping regions, a progressive image mosaic fusion algorithm in the form of a sine function is proposed. Finally, in order to make the progressive image mosaic fusion algorithm suitable for a variety of overlapping regions, this paper adds segmentation technology. The simulation experiment results show that the algorithm proposed in this paper is in good agreement with the spatial details and texture details of a high-resolution panchromatic image, and the time is shorter, which meets the real-time requirements. In addition, the algorithm proposed in this paper is effective in applications such as virtual reality and art image fusion.

Numerical results of Emden–Fowler boundary value problems with derivative dependence using the Bernstein collocation method

In this paper, we propose an efficient numerical technique based on the Bernstein polynomials for the numerical solution of the equivalent integral form of the derivative dependent Emden–Fowler boundary value problems which arises in various fields of applied mathematics, physical and chemical sciences. The Bernstein collocation method is used to convert the integral equation into a system of nonlinear equations. This system is then solved efficiently by suitable iterative method. The error analysis of the present method is discussed. The accuracy of the proposed method is examined by calculating the maximum absolute error and the $$L_{2}$$ error of four examples. The obtained numerical results are compared with the results obtained by the other known techniques.

Spatiotemporal Modelling using Integro-Difference Equations with Bivariate Stable Kernels

SummaryAn integro-difference equation can be represented as a hierarchical spatiotemporal dynamic model using appropriate parameterizations. The dynamics of the process defined by an integro-difference equation depends on the choice of a bivariate kernel distribution, where more flexible shapes generally result in more flexible models. Under a Bayesian modelling framework, we consider the use of the stable family of distributions for the kernel, as they are infinitely divisible and offer a variety of tail behaviours, orientations and skewness. Many of the attributes of the bivariate stable distribution are controlled by a measure, which we model using a flexible Bernstein polynomial basis prior. The method is the first attempt to incorporate non-Gaussian kernels in a two-dimensional integro-difference equation model and will be shown to improve prediction over the Gaussian kernel model for a data set of Pacific sea surface temperatures.

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

The q-Stirling numbers (polynomials) of the second kind have been investigated and applied in a variety of research subjects including, even, the q-analogue of Bernstein polynomials. The (p,q)-Stirling numbers (polynomials) of the second kind have been studied, particularly, in relation to combinatorics. In this paper, we aim to introduce new (p,q)-Stirling polynomials of the second kind which are shown to be fit for the (p,q)-analogue of Bernstein polynomials. We also present some interesting identities involving the (p,q)-binomial coefficients. We further discuss certain vanishing identities associated with the q-and (p,q)-Stirling polynomials of the second kind.

Fast and accurate evaluation of dual Bernstein polynomials

Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis, and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of these polynomials and their linear combinations. New simple recurrence relations of low order satisfied by dual Bernstein polynomials are given. In particular, a first-order non-homogeneous recurrence relation linking dual Bernstein and shifted Jacobi orthogonal polynomials has been obtained. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all n + 1 dual Bernstein polynomials of degree n with O(n) computational complexity.

On Multivariate Bernstein Polynomials

In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, fixed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions.

Numerical solution of multi-variable order fractional integro-differential equations using the Bernstein polynomials

Integro-differential equations are developed as models in enormous fields of engineering and science such as biological models, population growth, aerospace systems and industrial mathematics. In this work, we consider a general class of nonlinear fractional integro-differential equations with variable order derivative. We use the operational matrices based on the Bernstein polynomials to obtain numerical solution of this type of equations. By utilizing the operational matrices along with the Newton–Cotes collocation points, the problem under study is reduced to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, some examples are included to show the accuracy and validity of the proposed method.

A new approach for solving variable order differential equations based on Bernstein polynomials with Prabhakar function

A new approach for solving variable order differential equations based on Bernstein polynomials with Prabhakar function

Approximate multi-degree reduction of Q-B\xe9zier curves via generalized Bernstein polynomial functions

Quasi Bézier curves (or QB-curves, for short) possess the excellent geometric features of classical Bézier curves and also have good shape adjustability. In this paper, an algorithm for a multi-degree reduction of QB-curves based on L_{2} norm and by the analysis of geometric characteristics of QB-curves is constructed. The approximating approach for QB-curves of degree n+1 by degree m (mleq n) is also given. Secondly, by solving the linear equations under the constraints of C^{0} and C^{1} and without constraints, the explicit expression of the points of the approximating curve is obtained, which minimizes the error between the original curve and the approximating curve using the least square method. Some numerical examples of degree reduction under different constraints are given, and the corresponding errors are calculated as well. The results show that this method can be easily implemented, is highly precise and very effective.

Bayesian Nonparametric Monotone Regression.

In many applications there is interest in estimating the relation between a predictor and an outcome when the relation is known to be monotone or otherwise constrained due to the physical processes involved. We consider one such application-inferring time-resolved aerosol concentration from a low-cost differential pressure sensor. The objective is to estimate a monotone function and make inference on the scaled first derivative of the function. We proposed Bayesian nonparametric monotone regression which uses a Bernstein polynomial basis to construct the regression function and puts a Dirichlet process prior on the regression coefficients. The base measure of the Dirichlet process is a finite mixture of a mass point at zero and a truncated normal. This construction imposes monotonicity while clustering the basis functions. Clustering the basis functions reduces the parameter space and allows the estimated regression function to be linear. With the proposed approach we can make closed-formed inference on the derivative of the estimated function including full quantification of uncertainty. In a simulation study the proposed method performs similar to other monotone regression approaches when the true function is wavy but performs better when the true function is linear. We apply the method to estimate time-resolved aerosol concentration with a newly-developed portable aerosol monitor. The R package bnmr is made available to implement the method.

Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations and Their Numerical Solutions

In this paper, we investigate the existence and uniqueness of solution for a multi-term fractional integro-differential problem with nonlocal four-point fractional boundary conditions via the Caputo differentiation. We obtain operational matrix of Riemann–Liouville fractional integral operator of Bernstein polynomials and investigate the numerical solutions of the problem by using the collocation method. By appling these matrices fractional integro-differential equations convert to a linear system of equations. In this way, we give some examples to illustrate our results. The numerical method is computer oriented and produces very accurate and stable numerical results.

Generalized Gould-Hopper Based Fully Degenerate Central Bell Polynomials

In this paper, we first provide the generalized degenerate Gould-Hopper polynomials via thedegenerate exponential functions and then give various relations and formulas such as addition formula andexplicit identity. Moreover, we consider the generalized Gould-Hopper based degenerate central factorialnumbers of the second kind and present several identities and relationships. Furthermore, we introducethe generalized Gould-Hopper based fully degenerate central Bell polynomials and investigated multifariouscorrelations and formulas including summation formulas, derivation rule and correlations with the Stirlingnumbers of the first kind, the generalized Gould-Hopper based degenerate central factorial numbers of thesecond kind and the generalized degenerate Gould-Hopper polynomials. We then acquire some relations withthe degenerate Bernstein polynomials for the generalized Gould-Hopper based fully degenerate central Bellpolynomials. Finally, we consider the Gould-Hopper based fully degenerate Bernoulli, Euler and Genocchipolynomials and by utilizing these polynomials, we develop some representations for the generalized Gould-Hopper based fully degenerate central Bell polynomials.

Continuous-Time Look-Ahead Optimization of Energy Storage in Real-Time Balancing and Regulation Markets

This article proposes a novel continuous-time look-ahead optimization model for scheduling balancing energy and regulation capacity provided by energy storage (ES) devices and generating units in real-time markets. The proposed model implements the receding horizon control approach, where at each time, the operation of ES devices and generating units are co-optimized to minimize the total real-time system operation cost for providing the services over the control horizon in future. A continuous-time Gaussian process is developed to characterize the uncertainty and variability of load, which is used to update the load forecast over each control horizon. The proposed continuous-time problem is solved in a finite-dimensional function space spanned by Bernstein polynomials, which converts the problem into a solvable mixed-integer linear programming problem. The proposed model is implemented on the IEEE reliability test system. Numerical results show the optimal participation of ES in balancing and regulation markets considerably reduces the total real-time operation cost of the system.

On dual Bernstein polynomials and stochastic fractional integro‐differential equations

In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro‐differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.

An efficient solution of system of generalized Abel integral equations using Bernstein polynomials wavelet bases

This work introduces a direct method based on orthonormal Bernstein polynomials wavelet bases, to present a stable algorithm for numerical inversion of a system of generalized Abel integral equations. The application of all the currently existing numerical inversion methods was strictly limited to only one portion of the generalized Abel integral equations. The proposed method is quite accurate, and several numerical illustrations demonstrate the convergence and utilization of the proposed method compared to some of the preexisting numerical solution techniques. The permanence of the numerical result under the effect of small perturbation in input data has been examined, which is depicted with the use of numerical illustrations.

Variable selection for high-dimensional partly linear additive Cox model with application to Alzheimer's disease.

Variable selection has been discussed under many contexts and especially, a large literature has been established for the analysis of right-censored failure time data. In this article, we discuss an interval-censored failure time situation where there exist two sets of covariates with one being low-dimensional and having possible nonlinear effects and the other being high-dimensional. For the problem, we present a penalized estimation procedure for simultaneous variable selection and estimation, and in the method, Bernstein polynomials are used to approximate the involved nonlinear functions. Furthermore, for implementation, a coordinate-wise optimization algorithm, which can accommodate most commonly used penalty functions, is developed. A numerical study is performed for the evaluation of the proposed approach and suggests that it works well in practical situations. Finally the method is applied to an Alzheimer's disease study that motivated this investigation.

Evolutionary Algorithm Based Solution of Rössler Chaotic System Using Bernstein Polynomials

Chaotic systems have gained enormous research attention since the pioneering work of Lorenz. Rössler system stands among the extensively studied classical chaotic models. This paper proposes a technique based on Bernstein Polynomial Basis Function to convert the three-dimensional Rössler system of Ordinary Differential Equations (ODEs) into an error minimization problem. First, the properties of Bernstein Polynomials are applied to derive the fitness function of Rössler chaotic system. Second, in order to obtain the values of unknown Bernstein coefficients to optimize the fitness function, the problem is solved using two versatile algorithms from the family of Evolutionary Algorithms (EAs), Genetic Algorithm (GA) hybridized with Interior Point Algorithm (IPA) and Differential Algorithm (DE). For validity of the proposed technique, simulation results are provided which verify the global stability of error dynamics and provide accurate estimation of the desired parameters.

A new hybrid orthonormal Bernstein and improved block-pulse functions method for solving mathematical physics and engineering problems

In this paper, numerical solution of the linear second kind Fredholm integral equations is studied. These integral equations are widely used for solving many problems in mathematical physics and engineering. A new hybrid Bernstein and Improved Block-Pulse Functions (HBIBP) method is introduced and utilized to convert linear (nonlinear) second kind Fredholm integral equations into an algebraic equation. The new methodology is a combination of Bernstein polynomials (BPs) and improved block-pulse functions (IBPFs) on the interval [0, 1). Convergence analysis and numerical examples that illustrate the pertinent features of the method are presented.

A numerical approach for solving variable order differential equations using Bernstein polynomials

In this study, a numerical technique is applied to investigate solutions of linear variable order differential equations(VODEs) in fluid mechanics. We investigate variable order(VO) in the Caputo sense. Our aim is to apply Bernstein polynomials(Bps) as the basis functions and operational matrices. We calculate two types of operational matrices of Bps. Using these operational matrices the main equation can be transformed into a system of algebraic equations, which must then be solved to get approximate solutions. Some examples are provided to indicate the accuracy of the presented method.

NUMERICAL TREATMENT OF THE SPACE–TIME FRACTAL–FRACTIONAL MODEL OF NONLINEAR ADVECTION–DIFFUSION–REACTION EQUATION THROUGH THE BERNSTEIN POLYNOMIALS

In this paper, the nonlinear space–time fractal–fractional advection–diffusion–reaction equation is introduced and a highly accurate methodology is presented for its numerical solution. In the time direction, the fractal–fractional derivative in the Atangana–Riemann–Liouville concept is utilized whereas the fractional derivatives in the Caputo and Atangana–Baleanu–Caputo senses are mutually used in the space variable to define this new class of problems. The presented method utilizes the Bernstein polynomials (BPs) and their operational matrices of fractional and fractal–fractional derivatives (which are generated in this study). To this end, the unknown solution is expanded by the BP and is replaced in the equation. Then, the generated operational matrices and the collocation method are employed to generate a system of algebraic equations. Eventually, by solving this system a numerical solution is obtained for the problem. The validity of the designed method is investigated through three numerical examples.